A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric possess? The method is also applied to recognition of other polynomial integrals ...

We consider some special reductions of generic Darboux-Crum dressing formulas and of their difference versions. As a matter of fact we obtain some new formulas for Darboux-Pöschl-Teller potentials, their difference deformations and the related eigenfunctions.

A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.

A noncommutative version of the semi-discrete Toda equation is considered. A Lax pair and its Darboux transformations and binary Darboux transformations are found and they are used to construct two families of quasideterminant solutions.

In this paper, we construct two types of 7R closed single loop linkages by combining different factorizations of a general (non-vertical) Darboux motion. These factorizations are obtained by extensions of a factorization algorithm for a generic rational motion. The first type of 7R linkages has several one-dimensional configuration components and one of them corresponds to the Darboux motion. T...

We present algebraic construction of Darboux matrices for 1+1dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Bäcklund transformation, based on different λ-dependencies of the Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix form. We derive symme...

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvature in certain space forms is presented — in particular, a new form of Bryant...

The first Painlevé equation is expressed as a vector field on the three dimensional weighted projective space CP (3, 2, 4, 5). With the aid of the dynamical systems theory and the orbifold structure of CP (3, 2, 4, 5), a simple proof of the Painlevé property and the construction of the space of initial conditions are given. In particular, Painlevé’s transformation is geometrically derived, whic...

We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite genus curves. Our approach applies to complex-valued peri...